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Daily Archives: January 11, 2009

Can chaotic systems be predicted? I guess we first need to agree on exactly what a chaotic system is.

BusinessDictionary.com defines it as a “Complex system that shows sensitivity to initial conditions, such as an economy, a stockmarket, or weather. In such systems any uncertainty (no matter how small) in the beginning will produce rapidly escalating and compounding errors in the prediction of the system’s future behavior.”

It is hard to imagine a complex system that does not show sensitivity to initial conditions. If the follow-on statement is true, then there is little point to ever trying to model or predict the behavior of such a system because it is not predictable. But it is not hard to find counter-examples, even to the examples they provided. Meteorologists do a reasonable job predicting the weather; it depends on your standards of accuracy. Certainly they can predict fairly accurately the likelihood of a 90 degree day in January in Canada or anticipating the path of a tropical storm for the next 12 hours.

A less technical but perhaps more useful definition comes from membrane.com:“A chaotic system is one in which a tiny change can have a huge effect.”
That leads us toward a more practical definition for our purposes.

For the types of systems we normally model, I would propose yet another definition.A chaotic system is one in which it is likely that seemingly trivial changes in the initial conditions would cause significant changes in the predicted results, over the time frame being considered.

This definition, while not technically rigorous, acknowledges that most of us rarely have the opportunity or the need to deal in absolutes. We live in a world where the majority of decisions are made subjectively (“Joe has 20 years experience and he says…”) or with gross simplification (“Of course I can model that in a spreadsheet…”). In this world, being able to base a decision on a simulation model with better accuracy and objectivity can help realize tremendous savings, even if it is still only an approximation and only useful within specified parameters.

Can we accurately predict true chaotic systems? By strict definition clearly not. And even by my definition, there will be some systems that are just too chaotic to allow any predictions to be useful.

But can we provide useful predictions of most common systems, even those with some chaotic aspects? Absolutely yes. Every model is an approximation of a real or intended system. Part of our job as modelers is to ensure that the model is close enough to provide useful insight. A touch of chaos just makes that more interesting. 🙂